Question: Factor the following expression: $7$ $x^2$ $-18$ $x$ $-9$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(-9)} &=& -63 \\ {a} + {b} &=& & & {-18} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-63$ and add them together. Remember, since $-63$ is negative, one of the factors must be negative. The factors that add up to ${-18}$ will be your ${a}$ and ${b}$ When ${a}$ is ${3}$ and ${b}$ is ${-21}$ $ \begin{eqnarray} {ab} &=& ({3})({-21}) &=& -63 \\ {a} + {b} &=& {3} + {-21} &=& -18 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {7}x^2 +{3}x {-21}x {-9} $ Group the terms so that there is a common factor in each group: $ ({7}x^2 +{3}x) + ({-21}x {-9}) $ Factor out the common factors: $ x(7x + 3) - 3(7x + 3) $ Notice how $(7x + 3)$ has become a common factor. Factor this out to find the answer. $(7x + 3)(x - 3)$